Question

Solve the equation by factoring. \(x^2-8 x=-12\)

Short answer

The solutions for the equation \(x^2-8 x=-12\) are \(x=2, x=6\).

Step-by-step solution

Reconfigure into standard form

In this step, move all terms to the left side of the equation so as to form a quadratic equation. This is done by adding 12 to both sides to get the equation into the form \(x^2-8 x+12=0\)

Factoring

Next, factorize the quadratic equation \(x^2-8 x+12=0\). The equation can be factored into: \((x-2)(x-6)=0\) as -2 and -6 are the numbers which add up to -8 (the coefficient of x) and multiply to 12 (the constant term).

Solving for x

Now, set each factor to be equal to 0 and solve for x. So, \(x-2=0 \rightarrow x=2\) and \(x-6=0 \rightarrow x=6\). These are the roots or solutions for this quadratic equation.

Key concepts

Factoring Quadratics

Factoring quadratics is a method used to solve polynomial equations of the form \(ax^2 + bx + c = 0\). It's like reversing the process of multiplying out two binomials. The main idea is to express the quadratic equation as a product of two simple, linear factors.

• Start by rewriting the equation in standard form.
• Determine two numbers that multiply to the constant term \(c\) and add up to the linear coefficient \(b\).
• Rewrite the middle term using these two numbers, allowing you to group and factor by grouping.

For example, in the equation \(x^2-8x+12=0\), we needed numbers that multiply to 12 and add up to -8. These numbers are -2 and -6. Therefore, this quadratic can be factored into:
\((x - 2)(x - 6) = 0\).
Factoring is a straightforward approach when the equation can be easily rewritten using whole numbers. If that's not possible, other methods, like completing the square or the quadratic formula, may be used.

Solving Equations

Solving equations is about finding the values of variables that satisfy the equation. Once the quadratic equation is factored into two binomials, solving it becomes much simpler.
For the factored form \((x-2)(x-6)=0\), this highlights the 'Zero Product Property', which states that if a product of two factors equals zero, at least one of the factors must be zero.
These are the steps to solve the equation:

• Take each factor separately: \(x - 2 = 0\) and \(x - 6 = 0\).
• Solve each equation for \(x\).
• This yields the solutions: \(x = 2\) and \(x = 6\).

Breaking it down, solving a factored equation is direct. Each simple linear equation reflects a condition where the whole quadratic expression evaluates to zero, giving the roots of the equation.

Roots of Quadratic Equation

The roots of a quadratic equation are the values of \(x\) that make the equation true, which means finding the points where our equation crosses or touches the x-axis on a graph. These are also known as the solutions or zeros of the equation.
In our example, we found that the quadratic equation \(x^2-8x+12=0\) has roots \(x=2\) and \(x=6\).
Some properties of these roots include:

• The sum of the roots is equal to the negative of the linear coefficient divided by the leading coefficient \(-(b/a)\), which in our example is \(-(-8)/1 = 8\).
• The product of the roots is equal to the constant term divided by the leading coefficient \(c/a\), here it's \(12/1 = 12\).

Understanding these roots provides insights into the behavior of the quadratic function, including its graph, which is a parabola. Knowing the roots helps in graphing and applications where finding intersections is vital.